The following invention relates to a system and method for managing risk and, in particular, to a system and method for pricing default insurance for securities included in a basket of securities.
There are an increasing number of financial instruments that are used to manage the risk associated with various financial products. For example, many corporations and government agencies issue debt instruments that trade above or below standard benchmarks such as the London Interbank Offering Rate (LIBOR). Typically, a greater rate of interest above the LIBOR rate (i.e., spread) is paid to the corporate bond holder to compensate the owner against the risk of default by the issuer.
Various techniques may be used to protect the bond holder against this default risk. One such technique is to enter into a credit swap, also known as a default swap, in which the bond holder pays an insurance premium to an insurer in return for the right to “put” the bond to the insurer in case of a default event (for example, if the bond issuer fails to make coupon payments or files for bankruptcy). Thus, if a default occurs, the insurer must pay the bond holder an amount equal to the difference between the par value of the bond plus interest and the market value of the defaulted bond (typically well below par value). The insurance premium paid by the bond holder in a default swap is usually expressed as the difference between a risk-free benchmark yield (such as LIBOR) and the swap rate and is a good market indication of the credit-worthiness of the company issuing the bond.
Default insurance may also be purchased to protect against the default of any number of entities that are included in a basket of securities. Such a transaction, called a basket default swap, is similar to the default swap except that the insurance premium is paid by the buyer of insurance until a certain pre-specified number of entities in the basket have defaulted. For example, a buyer of “first to default” insurance in a basket of five entities receives default risk protection against a default of any one of the five issuers in the basket. The buyer may also purchase “second to default” insurance on the same basket in which case the buyer receives default protection in case a second issuer in the basket defaults. Similarly, the buyer may purchase insurance to protect against any other sequence of defaults that may occur. Basket default swaps are a common technique used to minimize the risk of default associated with owning Collateralized Bond Obligations (CBO's ), Collateralized Debt Obligations (CDO's ) and Collateralized Mortgage Obligations (CMO's ) as well as other types of portfolios.
In order to determine the insurance premium amount sufficient to compensate a counterparty for taking on the risk of default of a given security contained in a basket of securities, it is necessary to calculate the probability of default for each security in the basket. Generally, the calculated default risk depends on the possibility of one or multiple defaults in the basket occurring, any change in the market perceived probabilities of defaults as well as other factors including, but not limited to, changes in interest rates, exchange rates, credit ratings or regulations and counterparty risk.
Prior art techniques exist for determining the times of defaults for securities contained in a basket of securities. In one technique attributed to Duffie and Singleton, (see Darrell Duffie and Kenneth Singleton, Simulating Correlated Defaults, Graduate School of Business, Stanford University, May 1999), a barrier (that is a random variable) for each name in the basket is selected so that all the barriers are independent and conform to an exponential distribution. Next, a hazard rate (a random variable that reflects the instantaneous probability of default) is identified for each name in the basket. The time of default for a given name is then calculated by integrating the hazard rate over time and determining at what time the integral becomes greater than the barrier. Because this process for finding default times incorporates the non-deterministic nature of a default occurring (as reflected in the randomness of the hazard rate and barrier), this approach results in an accurate prediction of default times for the particular securities. Because, however, both the barriers and the hazard rates are stochastic processes, this technique requires that computationally intensive simulations (such as Monte Carlo simulations) be performed to solve for the default times. As a result, calculating default times under this prior art approach is slow which renders this approach impractical in many applications such as, for example, calculating insurance premiums for default insurance to support real-time markets trading in baskets of securities.
Another prior art approach for determining the times of defaults for securities contained in a basket of securities is attributed to David Li, (see David Li, On Default Correlation: A Copula Function Approach, RiskMetrics, April 2000 (hereinafter “Li”)), in which a copula function is directly used to determine the default times. The advantage of this approach is that because the copula function is deterministic, the need to run time-consuming simulations of hazard rates to calculate default times is eliminated and the computational speed of this approach is therefore increased. A drawback, however, of using this approach is that because the copula function does not explicitly take into account the correlation inherent in the hazard rates, the nature of the joint dynamics of spread movements of the entities in the basket is not accurately modeled, thereby resulting in decreased accuracy in the calculated default times.
Accordingly, it is desirable to provide a system and method for calculating the default times for entities in a basket of securities that is both accurate and computationally fast.